I dont know how to test any of the components for failure.
I have cleaned this board very well since these pictures were taken. There are 2 fine green wires running between chips, I have 'raised' them clear of components to check for shorts - none

Have gone over the board very carefully with magnifying glass both sides , cant see any obvious problems.

There are allot of 74 logic chips and I do have chips of the same value. I can safely remove chips and replace with new ones without damaging the board.

resoldering suspicious solderjoints.
In the picture of the solderside i've marked a bunch of solderjoints by highlighting them in red.
The solder at that joint does not look well and very aged or not
soldered solid in correct way.... i'd recomend to resolder that
joints by sucking off a portion of the solderlead and resolder
with a small portion of new solderlead.

2nd task:

Next i focus to the point of the shortcut....

i'd recommend to first "recap" the electrolytic capacitors.
I marked them in the picture by highlighting them yellow.
pay attention to the orientation and check for markuings of the "+"-side !
if needed mark that in a printout of the picture !

then it's recommended to replace the bridgerectifier diodes, that have straight
connection to higher current. I've marked those by enlighting them in violett.
Also here pay attention to the orientation ( printed at the board )
the arrow show to direction of a crossing line and that line marks
the position of the cathode.

the 2 chips enlighted by light purple seem to have been stressed by heat and should also get replaced.

and finally a really bad and ugly point:
the device spotted by the red arrow -
a coil marked as L3.... check for continutity .... it seems that coil has become fried....